= Solution
The <quadratic discrete predator-prey map> equals the identity plus the vector field of part (a), so its admissible <fixed points> are the same $O,E,P$. Their <Jacobian matrices> are the identity plus the corresponding continuous-time <Jacobian matrices>, but map stability requires <eigenvalues> inside the <unit circle> rather than the left half-plane.
At $O$ the multipliers are $1/a,0$: it is a sink for $a>1$ and a saddle for $a<1$. At $E$ they are $2-1/a$ and $(1-a)/b$. Therefore
$$
\boxed{E\text{ is a sink}\iff \frac13<a<1,\quad b>1-a.}
$$
For $a<1/3$, the prey multiplier is below $-1$: $E$ is a saddle when $b>1-a$ and a repeller when $b<1-a$. For $1/3<a<1$, $b<1-a$ makes it a saddle. The $+1$ crossings at $a=1$ and $a+b=1$ are the two <transcritical bifurcations> already encountered. The crossing at $a=1/3$ is a prey <period-doubling bifurcation>.
At positive coexistence,
$$
J(P)=\begin{pmatrix}1-b/a&-b\\c/b&1\end{pmatrix},\quad
T=2-b/a,\quad D=(1-2b)/a.
$$
For clarity, derive the relevant <Schur stability criterion> directly: setting $z=(1+s)/(1-s)$ in $z^2-Tz+D$ gives
$$
(1-s)^2p\left(\frac{1+s}{1-s}\right)
=(1-T+D)+2(1-D)s+(1+T+D)s^2.
$$
The open unit disk maps to $\operatorname{Re}s<0$. A real quadratic has both roots there precisely when all three displayed coefficients have the same positive sign. Since $1-T+D=c>0$, the coexistence <fixed point> is a sink precisely when
$$
\boxed{\frac{1-a}{2}<b<\min\left(1-a,a+\frac13\right).}
$$
This region exists only for $a>1/9$. In the coexistence triangle $a+b<1$, put $K=1+T+D=(3a+1-3b)/a$. If $K<0$, one root lies below $-1$ and the other in $(-1,1)$, so $P$ is a saddle. If $K>0$ but $D>1$, it is a repeller; if $K>0,D<1$, it is a sink. These signs give the complete local fixed-point partition, including real and complex multipliers.
The coexistence <period-doubling bifurcation> lies at $b=a+1/3$; its segment adjoining the sink region has $1/9<a<1/3$. The other multiplier there is $2-1/(3a)$ and lies strictly inside the <unit circle> on this segment. This coexistence flip is subcritical on that segment. To calculate its direction, normalize the $-1$ eigenvector by $q_f=(1,(2-b/a)/b)^T$ and the left eigenvector by $p_fq_f=1$. The symmetric quadratic derivative $\mathcal B$ is given explicitly in part (d). Eliminating the quadratic term from the one-dimensional <centre manifold> map gives the effective cubic flip coefficient
$$
c_f=\frac12p_f\mathcal B\left(q_f,(I-J)^{-1}\mathcal B(q_f,q_f)\right)
=-\frac9{(3a+1)^2(9a-1)}<0.
$$
For a scalar reduced map $u'=(-1+\delta)u+c_fu^3+\cdots$, its second iterate satisfies $u''-u=-2\delta u-2c_fu^3+\cdots$. Thus $u^2=-\delta/c_f>0$ on the stable-fixed-point side $\delta>0$, and the two-cycle multiplier is $1+4\delta+\cdots>1$. An unstable coexistence two-cycle meets the stable <fixed point> as $b$ increases to the flip curve. The other multiplier remains inside the unit circle. The complex unit-modulus crossing lies at $a+2b=1$, with $1/9<a<1$. The two curves meet at $(a,b)=(1/9,4/9)$ with a double multiplier $-1$. The prey flip and predator invasion curves meet at $(1/3,2/3)$ with multipliers $-1,+1$.
The sketches mark the sink regions and all these local thresholds. They are a local atlas, not a claim that the map has only these global attractors. Indeed the full nonnegative quadrant is not invariant: $x'=x(1-x-ay)/a$ becomes negative when $x+ay>1$. On the prey axis the <logistic map> already supplies new dynamics. Its nontrivial two-cycle, obtained by factoring $F^2(x)-x$ after removing the fixed-point factors, is
$$
x_\pm=\frac{1+a\pm\sqrt{(1-3a)(1+a)}}2,\qquad a<\frac13.
$$
Its longitudinal multiplier is $4+2/a-1/a^2$, so it is stable on the axis for $1/(1+\sqrt6)<a<1/3$. The transverse multiplier is $x_+x_-/b^2=a(1+a)/b^2$, making the cycle stable in the plane when additionally $b>\sqrt{a(1+a)}$. Further flips and more complicated logistic dynamics show why the differential equation's three global regions cannot be transferred unchanged to its unit-step map.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-65-map-regions.png]
{title=Local stability regions and fixed-point bifurcation curves of the quadratic predator–prey map}
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